Limits and Continuity of Functions

Abstract

In this section we extend the notion of the limit of a sequence to the concept of the limit of a function. Hereby we obtain a tool which enables us to investigate the behaviour of graphs of functions in the neighbourhood of chosen points. Moreover, limits of functions form the basis of one of the central themes in mathematical analysis, namely differentiation (Chap. 7). In order to derive certain differentiation formulae some elementary limits are needed, for instance, limits of trigonometric functions. The property of continuity of a function has far-reaching consequences, like, for instance, the intermediate value theorem, according to which a continuous function which changes sign in an interval has a zero. Not only does this theorem allow one to show the solvability of equations, it also provides numerical procedures to approximate the solutions. Further material on continuity can be found in Appendix C.